{"paper":{"title":"Proportion of cyclic matrices in maximal reducible matrix algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RT"],"primary_cat":"math.RA","authors_text":"Cheryl E. Praeger, Michael Giudici, Scott Brown, S.P. Glasby","submitted_at":"2014-05-22T05:01:27Z","abstract_excerpt":"Let ${\\rm M}(V)={\\rm M}(n,\\mathbb{F}_q)$ denote the algebra of $n\\times n$ matrices over $\\mathbb{F}_q$, and let ${\\rm M}(V)_U$ denote the (maximal reducible) subalgebra that normalizes a given $r$-dimensional subspace $U$ of $V=\\mathbb{F}_q^n$ where $0<r<n$. We prove that the density of non-cyclic matrices in ${\\rm M}(V)_U$ is at least $q^{-2}\\left(1+c_1q^{-1}\\right)$, and at most $q^{-2}\\left(1+c_2q^{-1}\\right)$, where $c_1$ and $c_2$ are constants independent of $n,r$, and $q$. The constants $c_1=-\\frac43$ and $c_2=\\frac{35}3$ suffice."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.6609","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}